Complexity of Positivstellensatz proofs for the knapsack

Grigoriev, Dima

doi:10.1007/s00037-001-8192-0

Cited by 104 publications

(124 citation statements)

References 15 publications

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“…By now, there's a large body of work that establishes lower bounds on SoS SDP for various average case problems. Beginning with the work of Grigoriev [Gri01a], a long line work have established tight lower bounds for random constraint satisfaction problems [Sch08,BCK15,KMOW17] and planted clique [MPW15, DM15, HKP15, RS15, SDP lower bounds for the planted clique problem were known for smaller degrees of sum-of-squares relaxations and for other SDP relaxations before; see the references therein for details.…”

Section: Related Workmentioning

confidence: 99%

The Power of Sum-of-Squares for Detecting Hidden Structures

Hopkins

Kothari²,

Potechin³

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

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We study planted problems-finding hidden structures in random noisy inputs-through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables.We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of size roughly n d . For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program.Using related ideas on SoS algorithms and low-degree matrix polynomials (and inspired by recent work on SoS and the planted clique problem [BHK + 16]), we prove new nearly-tight SoS lower bounds for the tensor and sparse principal component analysis problems. Our lower bounds are the first to suggest that improving upon the signal-to-noise ratios handled by existing polynomial-time algorithms for these problems may require subexponential time.

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Section: Related Workmentioning

confidence: 99%

The Power of Sum-of-Squares for Detecting Hidden Structures

Hopkins

Kothari²,

Potechin³

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

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“…This polynomial is nonnegative on all x ∈ {0, 1} n , and the induced matrix M g (x, y) = g(x ∧ y) is a submatrix of the slack matrix of the correlation polytope by Lemma 5. For odd n, Grigoriev [Gri01] shows that the sum-of-squares degree of g is n 2 (see also [Lau03]). Blekherman et al [BGP14] show that g even has high rational sum-of-squares degree: if the product pg has sos degree d, where p is an sos polynomials of degree r, then r + d ≥ 1 While we know that q is nonnegative on [n], we will not use this information.…”

Section: Quantum Query Complexity Lower Boundmentioning

confidence: 99%

Query Complexity in Expectation

Kaniewski

Lee

Wolf

2015

Automata, Languages, and Programming

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We study the query complexity of computing a function f : {0, 1}n → R + in expectation. This requires the algorithm on input x to output a nonnegative random variable whose expectation equals f (x), using as few queries to the input x as possible. We exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the sum-ofsquares degree, respectively. We observe that the quantum complexity can be unboundedly smaller than the classical complexity for some functions, but can be at most polynomially smaller for functions with range {0, 1}.These query complexities relate to (and are motivated by) the extension complexity of polytopes. The linear extension complexity of a polytope is characterized by the randomized communication complexity of computing its slack matrix in expectation, and the semidefinite (psd) extension complexity is characterized by the analogous quantum model. Since query complexity can be used to upper bound communication complexity of related functions, we can derive some upper bounds on psd extension complexity by constructing efficient quantum query algorithms. As an example we give an exponentially-close entrywise approximation of the slack matrix of the perfect matching polytope with psd-rank only 2. Finally, we show there is a precise sense in which randomized/quantum query complexity in expectation corresponds to the Sherali-Adams and Lasserre hierarchies, respectively.

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“…Most of the known lower bounds for the hierarchy originated in the works of Grigoriev [17,18] (also independently rediscovered later by Schoenebeck [36]). In [18] it is shown that random 3XOR or 3SAT instances cannot be solved by even Ω(n) rounds of SoS hierarchy.…”

Section: Introductionmentioning

confidence: 99%

“…Grigoriev [17] showed that n/2 levels of Lasserre are needed to prove that the polytope {x ∈ [0, 1] n | n i=1 x i = n/2 + 1/2} contains no integer point. A simplified proof can be found in [19].…”

Section: Introductionmentioning

confidence: 99%

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On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

Kurpisz

Leppänen

Mastrolilli

2015

Automata, Languages, and Programming

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The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems.In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n − 1. These problems are the hardest for the Lasserre hierarchy in this sense.

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Complexity of Positivstellensatz proofs for the knapsack

Cited by 104 publications

References 15 publications

The Power of Sum-of-Squares for Detecting Hidden Structures

The Power of Sum-of-Squares for Detecting Hidden Structures

Query Complexity in Expectation

On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

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